Geometric Algebra: Bivector, Comparison of Vector Algebra and Geometric Algebra, Paravector, Plane of Rotation, Spacetime Algebra

 
9781155703244: Geometric Algebra: Bivector, Comparison of Vector Algebra and Geometric Algebra, Paravector, Plane of Rotation, Spacetime Algebra

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 29. Chapters: Bivector, Comparison of vector algebra and geometric algebra, Paravector, Plane of rotation, Spacetime algebra, Algebra of physical space, Multivector, Blade, Pseudoscalar, Rotor. Excerpt: In mathematics, a bivector or 2-vector is a quantity in geometric algebra or exterior algebra that generalises the idea of a vector. If a scalar is considered a zero dimensional quantity, and a vector is a one dimensional quantity, then a bivector can be thought of as two dimensional. Bivectors have applications in many areas of mathematics and physics. They are related to complex numbers in two dimensions and to both pseudovectors and quaternions in three dimensions. They can be used to generate rotations in any dimension, and are a useful tool for classifying such rotations. They also are used in physics, tying together a number of otherwise unrelated quantities. Bivectors are generated by the exterior product on vectors - given two vectors a and b their exterior product is a bivector. But not all bivectors can be generated this way, and in higher dimensions a sum of exterior products is often needed. More precisely a bivector that requires only a single exterior product is simple; in two and three dimensions all bivectors are simple, but in higher dimensions this is not generally the case. The exterior product is antisymmetric, so negates the bivector, producing a rotation with the opposite sense, and is the zero bivector. Parallel plane segments with the same orientation and area corresponding to the same bivector .Geometrically, a simple bivector can be interpreted as an oriented plane segment, much as vectors can be thought of as directed line segments. Specifically for the bivector , its magnitude is the area of the parallelogram with edges a and b, its attitude that of any plane specified by a and b, and its orientation th...

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